A student performs an experiment to study the dependence of the time period (T) of a simple pendulum on its length. The length of the pendulum is changed each time. For every length, the time taken for 20 oscillations is recorded.

Note: The experiment is carried out in a school laboratory using the same bob.

The recorded observations are as follows:

Length (m)	Time taken for 20 oscillation (s)
0.40	25.4
0.60	31.0
0.90	38.0
1.00	40.2

(a) Calculate the time period for each length of the pendulum.

(b) Draw a graph between T² and the length of the pendulum.

(c) What conclusion can be drawn from the graph?

(d) Which physical parameter can be calculated using the slope of the above graph?

(e) If the same experiment is conducted on the Moon, would the graph remain the same? Give reason

(a) The time period T is the time taken for one oscillation and since the time given is for 20 oscillations, then time T is given by,

$\text T = \dfrac{\text {Time for 20 oscillations}}{20}$

$\begin{matrix} \text{Length (m)} & \text{Time taken (s)} & \text{Time Period} \\ 0.40 & 25.4 & \text T = \dfrac{\text {25.4}}{20} \\ & & \phantom{T \space} = 1.27 \text{ s} \\ 0.60 & 31.0 & \text T = \dfrac{\text {31.0}}{20} \\ & & \phantom{T \space} = 1.55 \text{ s} \\ 0.90 & 38.0 & \text T = \dfrac{\text {38.0}}{20} \\ & & \phantom{T \space} = 1.90 \text{ s} \\ 1.00 & 40.2 & \text T = \dfrac{\text {40.2}}{20} \\ & & \phantom{T \space} = 2.01 \text{ s} \\ \end{matrix}$

(b) The table below shows length (l) and T²:

The graph between T² and the length (l) of the pendulum is shown below :

(c) As the graph is a straight line, we can conclude that T² is directly proportional to the length (l) of the pendulum.

(d) From the relation

$\text T^2 = 4\text π^2\dfrac{\text l}{\text g}$

The slope of the T² vs l graph can be used to calculate the acceleration due to gravity (g).

(e) No, the graph would not remain the same on the Moon because the value of acceleration due to gravity (g) on the Moon is smaller than on Earth. Since $\text T^2 \propto \dfrac{1}{\text g}$, the time period would increase, changing the slope of the graph.